Influence techs are unlocked from the world map screen in the campaign and are not to be confused with the techs you research within a given mission. You earn influence by completing the primary objectives and king's objectives in missions. You can unlock an influence tech by clicking on the cup icon next to your prince's face on the world map. Each branch (Military, Production, Research) is labeled.
There is supposed to be just enough influence to complete two full branches in a campaign playthrough but a bug in the Infested Mountain level makes the it impossible to get credit for the king's objectives there meaning you can only complete one branch per playthrough. You should have enough influence to complete a branch after fully completing all objectives in all missions in the first three zones (6 missions) or right around completing the campaign if you don't bother going for the king's objectives. Note that you need to complete a mission after fully completing a branch for the achievement to unlock.